Method of reconstructing aspheric surface equations from measurements

ABSTRACT

Surface measurement data just provides the coordinates of an object surface without giving various parameters like the radius of curvature, conic constant, and deformation coefficients. In this paper, we propose a novel method for extracting the important parameters for the determination of unknown aspheric surface equations from the measurement of aspheric surfaces. The largest error between the original surface and the reconstructed surface in the theoretical case is shown to be about 8.6 nm. This fact implies that the new method is well suited for the reconstruction of unknown surface equations.

BACKGROUND OF INVENTION

Aspheric lens optics have received much attention in scientific, industrial, and military optical systems such as terahertz imaging, diagnostic systems, night-vision systems, and so on because they are able to remove spherical aberration and astigmatism for improving light gathering power and simplifying complex lens designs. For this reason, aspheric optics designs with parameters significantly affecting optical performance are treated as confidential information or are patented. Typically, aspheric surfaces fabricated according to surface designs are characterized by various evaluation methods. However, it is rare to reconstruct aspheric surface equations from surface measurements. In the present invention, a new method for reconstructing the aspheric surface equations from the measurement of unknown aspheric surfaces is disclosed.

SUMMARY OF INVENTION

A method of reconstructing aspheric surface equations in an information processing device from measurements, the method comprises steps for: obtaining z_(as)(r) as r varies from r_(i) to r_(f) in steps of r_(step) by measuring a given aspheric surface that is represented by an aspheric surface equation; finding an error curve δz_(e) for each R value while changing R from R_(i) to R_(f) in steps of R_(step) wherein K and deformation coefficients are set to 0, wherein the error curve δz_(e) is defined by a deviation of the aspheric surface equation from a quadratic term z_(s)(r) of the aspheric surface equation; finding the deformation coefficients from δz_(e) using an inverse matrix method; replacing R_(N) and a_(N) with each R value and the deformation coefficients found from the δz_(e) in z_(N), a curve determined by the measured coordinates; checking if a shape error, δz_(d), is within a predetermined threshold value; and determining R_(N) that is closest to the original R.

The aspheric surface equation may be given by

${{z(r)} = {\frac{\frac{r^{2}}{R}}{1 + \sqrt{1 - {\left( {1 + K} \right)\left( \frac{r}{R} \right)^{2}}}} + {\sum\limits_{i = 2}{a_{2i}r^{2i}}}}},$

where K is the conic constant, r is the lateral coordinate, R is the radius of curvature, and the a_(2i) values are the deformation coefficients.

The z_(as)(r_(i)), the δ_(s)(r), the δz_(e), the z_(N), and the δz_(d) may be given by

${z_{as}(r)} = {\frac{{cr}^{2}}{1 + \sqrt{1 - {\left( {1 + K} \right)c^{2}r^{2}}}} + {a_{4}r^{4}} + {a_{6}r^{6}} + {a_{8}r^{8}} + {a_{10}r^{10}}}$ ${z_{s}(r)} = \frac{{cr}^{2}}{1 + \sqrt{1 - {\left( {1 + K} \right)c^{2}r^{2}}}}$ δ z_(e) = z_(as) − z_(s) ${z_{N}(r)} = {{Re}\begin{bmatrix} {\frac{c_{N}r^{2}}{1 + \sqrt{1 - {\left( {1 + K} \right)c_{N}^{2}r^{2}}}} + {a_{N\; 4}r^{4}} +} \\ {{a_{N\; 6}r^{6}} + {a_{N\; 8}r^{8}} + {a_{N\; 10}r^{10}}} \end{bmatrix}}$ δ z_(d) = z_(as) − z_(N)

where c and c_(N) are the curvatures, which are the reciprocals of R and R_(N); z_(N) is the curve determined by the measured coordinate; a_(N4), . . . , a_(N10) are the deformation coefficients calculated with our method, and δz_(d) is the shape error, which is the deviation from the original curve z_(as).

In an embodiment of the present invention, the r_(i) may be 0, the r_(f) 35 mm, and the r_(step) about 3.5 mm.

The R_(i) may be 20 mm, the R_(f) 120 mm, and the R_(step) about 0.0001 mm.

The predetermined threshold value may be about 10 nm.

The method may further comprises steps for: setting the R and the R_(N) with the determined values; finding an error curve δz_(e) for each K value while changing K from K_(i) to K_(f) in steps of K_(step); finding the deformation coefficients from δz_(e) using an inverse matrix method; replacing K_(N) and a_(N) with each K value and the deformation coefficients found from the δz_(e) in z_(N), a curve determined by the measured coordinates; checking if a shape error, δz_(d), is within a predetermined threshold value; and determining K_(N) that is closest to the original K.

The K_(i) may be −1.0, the K_(f) 1.0, and the K_(step) about 0.001.

The inverse matrix method comprises steps for: representing the coordinate data (r_(i), z_(as)(r_(i))) by n+1 nth-order polynomial equations with polynomial coefficients, a₀, a₁, . . . , a_(n); writing the n+1 nth-order polynomial equations in a form of matrix multiplication, Xa=b, where X is a variable matrix, a is a coefficient vector, and b is a function vector; and calculating the polynomial coefficients, a₀, a₁, . . . , a_(n) using a=X⁻¹b.

The deformation coefficients may be obtained by

$\begin{bmatrix} a_{4} \\ a_{6} \\ a_{8} \\ a_{10} \end{bmatrix} = {{\begin{bmatrix} \left( r_{0} \right)^{4} & \left( r_{0} \right)^{6} & \left( r_{0} \right)^{8} & \left( r_{0} \right)^{10} \\ \left( r_{1} \right)^{4} & \left( r_{1} \right)^{6} & \left( r_{1} \right)^{8} & \left( r_{1} \right)^{10} \\ \vdots & \vdots & \vdots & \vdots \\ \left( r_{10} \right)^{4} & \left( r_{10} \right)^{6} & \left( r_{10} \right)^{8} & \left( r_{10} \right)^{10} \end{bmatrix}^{- 1}\begin{bmatrix} {\delta \; z_{e\; 0}} \\ {\delta \; z_{e\; 1}} \\ \vdots \\ {\delta \; z_{e\; 10}} \end{bmatrix}}.}$

The method according to the present invention provides a powerful method of reconstructing aspheric surface from measurements. And, further the method can be applied to aspheric lens optics easily.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 shows a flowchart illustrating steps for finding R in a method of reconstructing aspheric surface;

FIG. 2 shows a flowchart illustrating steps for finding K in a method of reconstructing aspheric surface;

FIG. 3 shows three different curves illustrating an original aspheric curve, a spherical curve, and an error curve;

FIGS. 4( a) to 4(d) each show procedures for finding a radius of curvature;

FIG. 5( a) shows a photograph of a setup for measuring surface; and

FIG. 5( b) shows a aspheric surface profile reconstructed by using an inverse matrix method.

DETAILED DESCRIPTION OF INVENTION

FIGS. 1 and 2 show flowcharts illustrating methods of reconstructing of unknown aspheric surface equations from measurement of aspheric surfaces.

A method of reconstructing aspheric surface equations in an information processing device from measurements, the method comprising steps for: obtaining z_(as)(r) as r varies from r, to r_(f) in steps of r_(step) by measuring a given aspheric surface that is represented by an aspheric surface equation (S10); finding an error curve δz_(e) for each R value while changing R from R_(i) to R_(f) in steps of R_(step) wherein K and deformation coefficients are set to 0, wherein the error curve δz_(e) is defined by a deviation of the aspheric surface equation from a quadratic term z_(s)(r) of the aspheric surface equation (S20); finding the deformation coefficients from δz_(e) using an inverse matrix method (S30); replacing R_(N) and a_(N) with each R value and the deformation coefficients found from the δz_(e) in z_(N), a curve determined by the measured coordinates (S40); checking if a shape error, δz_(d), is within a predetermined threshold value (S50); and determining R_(N) that is closest to the original R (S60).

The aspheric surface equation may be given by

${{z(r)} = {\frac{\frac{r^{2}}{R}}{1 + \sqrt{1 - {\left( {1 + K} \right)\left( \frac{r}{R} \right)^{2}}}} + {\sum\limits_{i = 2}{a_{2i}r^{2i}}}}},$

where K is the conic constant, r is the lateral coordinate, R is the radius of curvature, and the a_(2i), values are the deformation coefficients.

The z_(as)(r_(i)), the z_(s)(r), the δz_(e), the z_(N), and the δz_(d) may be given by

${z_{as}(r)} = {\frac{{cr}^{2}}{1 + \sqrt{1 - {\left( {1 + K} \right)c^{2}r^{2}}}} + {a_{4}r^{4}} + {a_{6}r^{6}} + {a_{8}r^{8}} + {a_{10}r^{10}}}$ ${z_{s}(r)} = \frac{{cr}^{2}}{1 + \sqrt{1 - {\left( {1 + K} \right)c^{2}r^{2}}}}$ δ z_(e) = z_(as) − z_(s) ${z_{N}(r)} = {{Re}\begin{bmatrix} {\frac{c_{N}r^{2}}{1 + \sqrt{1 - {\left( {1 + K} \right)c_{N}^{2}r^{2}}}} + {a_{N\; 4}r^{4}} +} \\ {{a_{N\; 6}r^{6}} + {a_{N\; 8}r^{8}} + {a_{N\; 10}r^{10}}} \end{bmatrix}}$ δ z_(d) = z_(as) − z_(N)

where c and c_(N) are the curvatures, which are the reciprocals of R and R_(N); z_(N) is the curve determined by the measured coordinate; a_(N4), . . . , a_(N10) are the deformation coefficients calculated with our method, and δz_(d) is the shape error, which is the deviation from the original curve z_(as).

In an embodiment of the present invention, the r, may be 0, the r_(f) 35 mm, and the r_(step) about 3.5 mm. The R_(i) may be 20 mm, the R_(f) 120 mm, and the R_(step) about 0.0001 mm. The predetermined threshold value may be about 10 nm.

In addition to the above steps for finding R, in order to find K, the method may further comprises steps for: setting the R and the R_(N) with the determined values (S70); finding an error curve δz_(e) for each K value while changing K from K_(i) to K_(f) in steps of K_(step) (S80); finding the deformation coefficients from δz_(e) using an inverse matrix method (S90); replacing K_(N) and a_(N) with each K value and the deformation coefficients found from the δz_(e) in z_(N), a curve determined by the measured coordinates (S100); checking if a shape error, δz_(d), is within a predetermined threshold value (S110); and determining K_(N) that is closest to the original K (S120).

The K_(i) may be −1.0, the K_(f) 1.0, and the K_(step) about 0.001.

The inverse matrix method comprises steps for: representing the coordinate data (r_(i), z_(as)(r_(i))) by n+1 nth-order polynomial equations with polynomial coefficients, a₀, a₁, . . . , a_(n); writing the n+1 nth-order polynomial equations in a form of matrix multiplication, Xa=b, where X is a variable matrix, a is a coefficient vector, and b is a function vector; and calculating the polynomial coefficients, a₀, a₁, . . . , a_(n) using a=X⁻¹b.

The deformation coefficients may be obtained by

$\begin{bmatrix} a_{4} \\ a_{6} \\ a_{8} \\ a_{10} \end{bmatrix} = {{\begin{bmatrix} \left( r_{0} \right)^{4} & \left( r_{0} \right)^{6} & \left( r_{0} \right)^{8} & \left( r_{0} \right)^{10} \\ \left( r_{1} \right)^{4} & \left( r_{1} \right)^{6} & \left( r_{1} \right)^{8} & \left( r_{1} \right)^{10} \\ \vdots & \vdots & \vdots & \vdots \\ \left( r_{10} \right)^{4} & \left( r_{10} \right)^{6} & \left( r_{10} \right)^{8} & \left( r_{10} \right)^{10} \end{bmatrix}^{- 1}\begin{bmatrix} {\delta \; z_{e\; 0}} \\ {\delta \; z_{e\; 1}} \\ \vdots \\ {\delta \; z_{e\; 10}} \end{bmatrix}}.}$

In general, arbitrary curves including two-dimensional shapes are given by data in the form of various coordinates, (x₀, y₀), (x₁, y₁), . . . , (x_(n), y_(n)). This data can be represented by an nth-order polynomial equation, as follows:

y=a ₀ +a ₁ x+a ₂ x ² + . . . +a _(n) x ^(n)  (1)

where a₀, a₁, . . . , a_(n) are the polynomial coefficients. Substituting each of the n+1 (x, y) pairs into Eq. (1) yields the following n+1 equations:

$\begin{matrix} {{{a_{0} + {a_{1}x_{0}} + {a_{2}\left( x_{0} \right)}^{2} + \ldots + {a_{n}\left( x_{0} \right)}^{n}} = y_{0}}{{a_{0} + {a_{1}x_{1}} + {a_{2}\left( x_{1} \right)}^{2} + \ldots + {a_{n}\left( x_{1} \right)}^{n}} = y_{1}}\vdots {{a_{0} + {a_{1}x_{n}} + {a_{2}\left( x_{n} \right)}^{2} + \ldots + {a_{n}\left( x_{n} \right)}^{n}} = y_{n}}} & (2) \end{matrix}$

Equation (2) is a system of nth-order equations having the unknown quantities a₀, a₁, . . . , a_(n). Here, however, Eq. (2) can be considered a system of linear equations with respect to the a_(i) coefficients (where i=0, . . . , n) because the (x, y) values can be determined through measurements. Therefore, the equation can be written in the form of matrix multiplication, Xa=b, where X is the variable matrix, a is the coefficient vector, and b is the function vector. The result, represented as a matrix calculation, is as follows:

$\begin{matrix} {{\begin{bmatrix} 1 & x_{0} & \left( x_{0} \right)^{2} & \ldots & \left( x_{0} \right)^{n} \\ 1 & x_{1} & \left( x_{1} \right)^{2} & \ldots & \left( x_{1} \right)^{n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_{n} & \left( x_{n} \right)^{2} & \ldots & \left( x_{n} \right)^{n} \end{bmatrix}\begin{bmatrix} a_{0} \\ a_{1} \\ \vdots \\ a_{n} \end{bmatrix}} = \begin{bmatrix} y_{0} \\ y_{1} \\ \vdots \\ y_{n} \end{bmatrix}} & (3) \end{matrix}$

We want to solve Eq. (3) for the elements of the vector a. So, a can be found by multiplying the inverse matrix X⁻¹ of X at either side, as follows:

X ⁻¹ Xa=X ⁻¹ b

a=X ⁻¹ b  (4)

In order for it be possible to calculate the inverse of the matrix X (that is, for X to be invertible), det(X) must not be equal to zero. In practice, if a is found by Eq. (4), the original equation can be reconstructed from the coefficients in the vector a. If we measure the curve given by a certain equation, the measured values will be y values for the corresponding x values. An important point here is that when we do not know the equation of the curve, the measured value will be y, which contains the coefficients multiplied by suitable powers of the corresponding x value. So, the coefficients become the elements of the vector a extracted from the calculation in Eq. (4). We refer to this method as the inverse matrix method.

In general, an aspheric surface equation⁷ is given by

$\begin{matrix} {{z(r)} = {\frac{\frac{r^{2}}{R}}{1 + \sqrt{1 - {\left( {1 + K} \right)\left( \frac{r}{R} \right)^{2}}}} + {\sum\limits_{i = 2}{a_{2i}r^{2i}}}}} & (5) \end{matrix}$

where K is the conic constant, r is the lateral coordinate, R is the radius of curvature, and the a_(2i) values are the deformation coefficients. To verify the validity of the inverse matrix method, let us suppose an arbitrary aspheric equation z_(as), which has a radius of curvature at the vertex is R=89 mm, a diameter of φ5=70 mm, a conic constant of K=0, and deformation coefficients of a₄=−5×10⁻⁹, a₆=−2.075×10⁻⁹, a₈=7.65625×10⁻¹³, and a₁₀=−1.67481×10⁻¹⁶. Now, we must reconstruct the aspheric surface equation from the coordinate data (r, z_(as)) of Eq. (5) with the known parameters. When K=0 and a_(2i)=0, Eq. (5) includes a spherical equation that can be considered as the basic form of the aspheric surface equation. So R, the radius of the circle, must be found. In order to find R, we must calculate the error curve δz_(e)(r)=z_(as)−z_(s), where z_(as) is the entire equation itself and z_(s) is just the quadratic term in Eq. (5), neglecting the higher-order terms. FIG. 3 shows (a) the original aspheric curve z_(as), (b) the spherical curve z_(s), and (c) the error curve δz_(e), which represents the difference between z_(as) and z_(s). The deformation coefficients can be found from the error curve by using the inverse matrix method.

As rvaries from 0 to 35 mm in steps of 3.5 mm, 11 data points are obtained from δz_(e)(r). Here, as there are 11 r values, 11 deformation coefficients are also obtained. Among these coefficients, we are interested in the 4th-, 6th-, 8th-, and 10th-order coefficients due to Eq. (5). Therefore, others coefficients, like the constant and the odd-order coefficients, are omitted in the process of matrix calculation of the error curve. This calculation of the error curve is also in the form of Xa=b like Eq. (3) and is given by Eq. (6). In Eq. (6), the X matrix is 11×4, and the a vector is 4×1, so the b vector must be 11×1. In order to find the coefficient vector a, both sides of the equation are multiplied by the inverse matrix X⁻¹, and since X⁻¹ is 4×11, the product of X⁻¹ and b will be 4×1. Therefore, the matrix calculation works because the result equals a, as shown by Eq. (7).

$\begin{matrix} {{\begin{bmatrix} \left( r_{0} \right)^{4} & \left( r_{0} \right)^{6} & \left( r_{0} \right)^{8} & \left( r_{0} \right)^{10} \\ \left( r_{1} \right)^{4} & \left( r_{1} \right)^{6} & \left( r_{1} \right)^{8} & \left( r_{1} \right)^{10} \\ \vdots & \vdots & \vdots & \vdots \\ \left( r_{10} \right)^{4} & \left( r_{10} \right)^{6} & \left( r_{10} \right)^{8} & \left( r_{10} \right)^{10} \end{bmatrix}\begin{bmatrix} a_{4} \\ a_{6} \\ a_{8} \\ a_{10} \end{bmatrix}} = \begin{bmatrix} {\delta \; z_{e\; 0}} \\ {\delta \; z_{e\; 1}} \\ \vdots \\ {\delta \; z_{e\; 10}} \end{bmatrix}} & (6) \\ {\begin{bmatrix} a_{4} \\ a_{6} \\ a_{8} \\ a_{10} \end{bmatrix} = {\begin{bmatrix} \left( r_{0} \right)^{4} & \left( r_{0} \right)^{6} & \left( r_{0} \right)^{8} & \left( r_{0} \right)^{10} \\ \left( r_{1} \right)^{4} & \left( r_{1} \right)^{6} & \left( r_{1} \right)^{8} & \left( r_{1} \right)^{10} \\ \vdots & \vdots & \vdots & \vdots \\ \left( r_{10} \right)^{4} & \left( r_{10} \right)^{6} & \left( r_{10} \right)^{8} & \left( r_{10} \right)^{10} \end{bmatrix}^{- 1}\begin{bmatrix} {\delta \; z_{e\; 0}} \\ {\delta \; z_{e\; 1}} \\ \vdots \\ {\delta \; z_{e\; 10}} \end{bmatrix}}} & (7) \end{matrix}$

In reality, we only know the coordinate data (r, z_(as)) from the surface measurement and not R, K, and the a coefficient values. The following equations are given to find R, K, and the a coefficients.

$\begin{matrix} {{{z_{as}(r)} = {\frac{{cr}^{2}}{1 + \sqrt{1 - {\left( {1 + K} \right)c^{2}r^{2}}}} + {a_{4}r^{4}} + {a_{6}r^{6}} + {a_{8}r^{8}} + {a_{10}r^{10}}}}{{z_{s}(r)} = \frac{{cr}^{2}}{1 + \sqrt{1 - {\left( {1 + K} \right)c^{2}r^{2}}}}}{{\delta \; z_{e}} = {z_{as} - z_{s}}}{{z_{N}(r)} = {{Re}\begin{bmatrix} {\frac{c_{N}r^{2}}{1 + \sqrt{1 - {\left( {1 + K} \right)c_{N}^{2}r^{2}}}} + {a_{N\; 4}r^{4}} +} \\ {{a_{N\; 6}r^{6}} + {a_{N\; 8}r^{8}} + {a_{N\; 10}r^{10}}} \end{bmatrix}}}{{\delta \; z_{d}} = {{z_{as} - z_{N}}}}} & (8) \end{matrix}$

where c and c_(N) are the curvatures, which are the reciprocals of R and R_(N); z_(N) is the curve determined by the measured coordinate; a_(N4), . . . , a_(N10) are the deformation coefficients calculated with our method, and δz_(d) is the shape error, which is the deviation from the original curve z_(as). The procedure for finding the parameters is as follows:

(1) Obtain z_(as) through measurements as r varies from 0 to 35 mm in steps of 3.5 mm.

(2) Change R from 20 to 120 mm in steps of 0.0001 mm, where K and a₄, . . . , a₁₀ are set to 0.

(3) Find the error curve δz_(e) for each R value.

(4) Find the deformation coefficients from δz_(e) using the inverse matrix method.

(5) In z_(N), replace R_(N) and a_(N) with each R value and the deformation coefficients found from δz_(e).

(6) Check if δz_(d) is within a desired threshold value.

This procedure was implemented in an information processing unit such as a computer, using a well-known MATLAB® program and was repeated iteratively until the δz_(d) value fell below some threshold value. In this case, the threshold value was 1×10⁻⁵ mm (10 nm), and then the relevant R_(N) was 89.0013 mm. FIGS. 4( a) to 4(d) each shows the procedure for finding the R_(N) closest to the original R, more specifically, for finding the radius of curvature R_(N), where the red line is z_(as), the cyan line is z_(N), and the green line is z_(d), in which (a) R_(N)=20.0000 mm, (b) R_(N)=60.0000 mm, (c) R_(N)=89.0013 mm, and (d) R_(N)=120.0000 mm. As R_(N) approaches the original R, δz_(d) goes to 0. Here, the minimized δz_(d) was obtained when R_(N)=89.0013 mm as shown in FIG. 4( c). After that, R_(N) increased further, and δz_(d) increased again. Table 1 shows δz_(d) for various rvalues when R_(N) is 89.0013 mm. Even the largest δz_(d) is no larger than about 9.77 nm at r=31.5 mm, so this result is quite satisfactory. One thing to keep in mind is that the coordinate data of z_(as) will correspond to the measurement data of an unknown aspheric lens surface later, and it will become a reference used to find R.

TABLE 1 Minimized δz_(d) values found for R_(N) = 89.0013 mm R_(N) (mm) r (mm) δz_(d) (mm) 89.0013 0 0.0000 3.5 9.2793 × 10⁻⁷ 7 2.8688 × 10⁻⁶ 10.5 3.8795 × 10⁻⁶ 14 2.3828 × 10⁻⁶ 17.5 1.2827 × 10⁻⁶ 21 4.3100 × 10⁻⁶ 24.5 2.8575 × 10⁻⁶ 28 4.1517 × 10⁻⁶ 31.5 9.7711 × 10⁻⁶ 35 4.6779 × 10⁻⁶

The deformation coefficients extracted from the inverse matrix method (7) for R_(N)=89.0013 mm are compared with the original one in Table 2. The extracted

TABLE 2 Comparison between original and extracted deformation coefficients Deformation coefficient Original Extracted a₄     −5 × 10⁻⁹  −4.465 × 10⁻⁹ a₆  −2.075 × 10⁻⁹  −2.076 × 10⁻⁹ a₈  7.65625 × 10⁻¹³  7.6632 × 10⁻¹³  a₁₀ −1.67481 × 10⁻¹⁶ −1.6763 × 10⁻¹⁶ where coefficients are very similar to the original ones. This suggests that the inverse matrix method is quite effective.

In the case where K=−0.8 in Eq. (5) rather than K=0, we were able to find R, K, and a using the inverse matrix method. In the same way, when K=0, Rwas found to be 88.998. In this case, the R value (88.998) differs from its value in the previous case (89.0013) due to the difference in K. Now, K can be also determined in a way that is similar to the way that R was found. To find K, we applied R=88.998 to Eq. (8) and changed K from −1 to 1 in steps of 0.001, and then examined the value of K that occurred for shape error values less than δz_(d)=1×10⁻⁵ mm. The resultant K was −0.732, and at that time, the largest δz_(d) was 8.6152×10⁻⁶ mm (about 8.6 nm). Finally, the deformation coefficients were a₄=−1.7652×10⁻⁸, a₆=−2.0746×10⁻⁹, a₈=7.6537×10⁻¹³, and a₁₀=−1.6752×10⁻¹⁶. Although the extracted K and the deformation coefficients are somewhat different than the original ones, the result is satisfactory because the largest shape error δz_(d) is no larger than about 8.6 nm.

As a last step we applied the inverse matrix method to a real situation. A lens surface with unknown parameters was measured by a CONTURA G2® (Karl Zeiss®) which is a 3-dimensional measurement device. The parameters extracted from the measured coordinates using the inverse matrix method were R=121.33 mm, K=0, a₄=1.4193×10⁻⁶, a₆=−3.4706×10⁻⁹, a₈=3.8349×10⁻¹², and a₁₀=−1.5313×10⁻¹⁵. Then the largest shape error δz_(d) between the measured coordinate data and the data from the reconstructed equation was 2.328×10⁻³ mm (about 2.3 μm). In this case, a major cause of δz_(d) being higher than that in the theoretical case may be the precision limitation of the measurement equipment, which has a repeatability precision of 1.8 μm, and the uncertainty of the measurement⁷, which is the doubt that exists about the result of any measurement. Accordingly, these limitations can be overcome if equipment with higher precision is used and if a large number of measurements are performed. FIG. 5 shows the measurement of the aspheric lens surface and the surface profile realized by using the reconstructed surface equation found from the inverse matrix method. The procedure for finding R, K, and the deformation coefficients of the unknown aspheric lens surface is shown with (a) photograph of the setup for the surface measurement and (b) the aspheric surface profile reconstructed by using the inverse matrix method.

In the above steps, all the programming codes, intermediate results, and necessary information for performing them may be stored in a memory device of the information processing device. In other words, the memory of the computer may be modified with the results of the steps. Of course, the performing of the steps is not limited to MATLAB®. It is just a choice out of many appropriate hardware or software means that can process the inventive steps.

In conclusion, a new method using the inverse matrix method is disclosed. From the results of this study, it may be concluded that the inverse matrix method is a powerful means for reconstructing the unknown equations of various aspheric surfaces, and it is possible for this technique to be applied to other fields. 

What is claimed is:
 1. A method of reconstructing aspheric surface equations in an information processing device from measurements, the method comprising steps for: obtaining z_(as)(r) as a lateral coordinate r varies from r_(i) to r_(f) in steps of r_(step) by measuring a given aspheric surface that is represented by an aspheric surface equation; finding an error curve δz_(e) for each radius of curvature R value while changing R from R_(i) to R_(f) in steps of R_(step) wherein a conic constant K and deformation coefficients are set to 0, wherein the error curve δz_(e) is defined by a deviation of the aspheric surface equation from a quadratic term z_(s)(r) of the aspheric surface equation; finding the deformation coefficients from δz_(e) using an inverse matrix method; replacing R_(N) and a_(N) with each R value and the deformation coefficients found from the δz_(e) in z_(N), a curve determined by the measured coordinates; checking if a shape error, δz_(d), is within a predetermined threshold value; and determining R_(N) that is closest to the original R.
 2. The method of claim 1, wherein the aspheric surface equation is given by ${{z(r)} = {\frac{\frac{r^{2}}{R}}{1 + \sqrt{1 - {\left( {1 + K} \right)\left( \frac{r}{R} \right)^{2}}}} + {\sum\limits_{i = 2}{a_{2i}r^{2i}}}}},$ where K is the conic constant, r is the lateral coordinate, R is the radius of curvature, and the a_(2i) values are the deformation coefficients.
 3. The method of claim 2, wherein the z_(as)(r_(i)), the z_(s)(r), the δz_(e), the z_(N), and the δz_(d) are given by ${z_{as}(r)} = {\frac{{cr}^{2}}{1 + \sqrt{1 - {\left( {1 + K} \right)c^{2}r^{2}}}} + {a_{4}r^{4}} + {a_{6}r^{6}} + {a_{8}r^{8}} + {a_{10}r^{10}}}$ ${z_{s}(r)} = \frac{{cr}^{2}}{1 + \sqrt{1 - {\left( {1 + K} \right)c^{2}r^{2}}}}$ δ z_(e) = z_(as) − z_(s) ${z_{N}(r)} = {{Re}\begin{bmatrix} {\frac{c_{N}r^{2}}{1 + \sqrt{1 - {\left( {1 + K} \right)c_{N}^{2}r^{2}}}} + {a_{N\; 4}r^{4}} +} \\ {{a_{N\; 6}r^{6}} + {a_{N\; 8}r^{8}} + {a_{N\; 10}r^{10}}} \end{bmatrix}}$ δ z_(d) = z_(as) − z_(N) where c and c_(N) are the curvatures, which are the reciprocals of R and R_(N); z_(N) is the curve determined by the measured coordinate; a_(N4), . . . , a_(N10) are the deformation coefficients calculated with our method, and δz_(d) is the shape error, which is the deviation from the original curve z_(as).
 4. The method of claim 3, wherein the r_(i) is 0, the r_(f) is 35 mm, and the r_(step) is about 3.5 mm.
 5. The method of claim 3, wherein the R_(i) is 20 mm, the R_(f) is 120 mm, and the R_(step) is about 0.0001 mm.
 6. The method of claim 3, wherein the predetermined threshold value is about 10 nm.
 7. The method of claim 3, further comprising steps for: setting the R and the R_(N) with the determined values; finding an error curve δz_(e) for each K value while changing K from K_(i) to K_(f) in steps of K_(step); finding the deformation coefficients from δz_(e) using an inverse matrix method; replacing K_(N) and a_(N) with each K value and the deformation coefficients found from the δz_(e) in z_(N), a curve determined by the measured coordinates; checking if a shape error, δz_(d), is within a predetermined threshold value; and determining K_(N) that is closest to the original K.
 8. The method of claim 7, wherein the K_(i) is −1.0, the K_(f) is 1.0, and the K_(step) is about 0.001.
 9. The method of claim 3, wherein the inverse matrix method comprises steps for: representing the coordinate data (r_(i), z_(as)(r_(i))) by n+1 nth-order polynomial equations with polynomial coefficients, a₀, a₁, . . . , a_(n); writing the n+1 nth-order polynomial equations in a form of matrix multiplication, Xa=b, where X is a variable matrix, a is a coefficient vector, and b is a function vector; and calculating the polynomial coefficients, a₀, a₁, . . . , a_(n) using a=X⁻¹ b.
 10. The method of claim 9, wherein the deformation coefficients are obtained by $\begin{bmatrix} a_{4} \\ a_{6} \\ a_{8} \\ a_{10} \end{bmatrix} = {{\begin{bmatrix} \left( r_{0} \right)^{4} & \left( r_{0} \right)^{6} & \left( r_{0} \right)^{8} & \left( r_{0} \right)^{10} \\ \left( r_{1} \right)^{4} & \left( r_{1} \right)^{6} & \left( r_{1} \right)^{8} & \left( r_{1} \right)^{10} \\ \vdots & \vdots & \vdots & \vdots \\ \left( r_{10} \right)^{4} & \left( r_{10} \right)^{6} & \left( r_{10} \right)^{8} & \left( r_{10} \right)^{10} \end{bmatrix}^{- 1}\begin{bmatrix} {\delta \; z_{e\; 0}} \\ {\delta \; z_{e\; 1}} \\ \vdots \\ {\delta \; z_{e\; 10}} \end{bmatrix}}.}$ 